A Framework for Reliability Approximation of Multi-State Weighted k-out-of-n Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The multi-state <formula formulatype="inline"><tex Notation="TeX">k</tex> </formula>-out-of-<formula formulatype="inline"><tex Notation="TeX">n</tex> </formula> system model finds wide applications in industry, and has been extensively studied in recent years. This model has also been generalized to the multi-state weighted <formula formulatype="inline"><tex Notation="TeX">k</tex> </formula>-out-of-<formula formulatype="inline"><tex Notation="TeX">n</tex> </formula> system model. Recursive methods, and universal generating functions (UGF) are two primary algorithms for exact performance evaluation of multi-state <formula formulatype="inline"><tex Notation="TeX">k</tex></formula>-out-of-<formula formulatype="inline"><tex Notation="TeX">n</tex></formula> systems. However the computational burden becomes the crucial factor when there is a “dimension damnation” problem caused by the increase in the number of components in the system, and the number of possible states a component may be in. In situations wherein exact values of system reliability are not necessary, we may use more efficient algorithms to approximate system reliability. In this paper, we develop a comprehensive framework for reliability approximation of multi-state weighted <formula formulatype="inline"><tex Notation="TeX">k</tex> </formula>-out-of-<formula formulatype="inline"><tex Notation="TeX">n</tex> </formula> systems. Two fuzzy based multi-state weighted <formula formulatype="inline"> <tex Notation="TeX">k</tex></formula>-out-of-<formula formulatype="inline"> <tex Notation="TeX">n</tex></formula> system models are defined. Procedures for building these two models from the conventional models are also introduced. The fuzzy recursive methods, and fuzzy UGF techniques are developed to evaluate such systems. The clustering technique, and curve fitting method are used to determine the fuzzy weights, and probabilities of states in the models. </para>